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Idea

A topological space (or more generally, a convergence space) is Hausdorff if convergence is unique. The concept can also be defined for locales (see Definition 3 below) and categorified (see Beyond topological spaces below). A Hausdorff space is often called $T_2$, since this condition came second in the original list of four separation axioms (there are more now) satisfied by metric spaces.

the main separation axioms

number	name	statement	reformulation
$T_0$	Kolmogorov	given two distinct points, at least one of them has an open neighbourhood not containing the other point	every irreducible closed subset is the closure of at most one point
$T_1$		given two distinct points, both have an open neighbourhood not containing the other point	all points are closed
$T_2$	Hausdorff	given two distinct points, they have disjoint open neighbourhoods	the diagonal is a closed map
$T_{\gt 2}$		$T_1$ and…	all points are closed and…
$T_3$	regular Hausdorff	…given a point and a closed subset not containing it, they have disjoint open neighbourhoods	…every neighbourhood of a point contains the closure of an open neighbourhood
$T_4$	normal Hausdorff	…given two disjoint closed subsets, they have disjoint open neighbourhoods	…every neighbourhood of a closed set also contains the closure of an open neighbourhood … every pair of disjoint closed subsets is separated by an Urysohn function

Hausdorff spaces are a kind of nice topological space; they do not form a particularly nice category of spaces themselves, but many such nice categories consist of only Hausdorff spaces. In fact, Felix Hausdorff's original definition of ‘topological space’ actually required the space to be Hausdorff, hence the name. Certainly homotopy theory (up to weak homotopy equivalence) needs only Hausdorff spaces. It is also common in analysis to assume that all spaces encountered are Hausdorff; if necessary, this can be arranged since every space has a Hausdorff quotient (in fact, the Hausdorff spaces form a reflective subcategory of Top), although usually an easier method is available than this sledgehammer.

Definitions

There are many equivalent ways of characterizing a space $S$ as Hausdorff. The traditional definition is this:

That is, any two distinct points can be separated by open neighbourhoods, and it is simply a mundane way of saying that $\ne$ is open in the product topology on $S \times S$.

Here is a classically equivalent definition that is more suitable for constructive mathematics:

This is the mundane way of saying that $=$ is closed in $S \times S$.

Another way of saying this, which makes sense also for locales, is the following:

This way of stating the definition generalizes to topos theory and thus to many other contexts; but it is not always a faithful generalization of the classical notion for topological spaces. See Beyond topological spaces below for more.

Here is an equivalent definition (constructively equivalent to Definition 2) that makes sense for arbitrary convergence spaces or preconvergence generalised filter spaces:

Or equivalently,

\begin{definition} Given a net (or equivalently, a proper filter) $F$ in $S$, the set of all limits of $F$ is a subsingleton. \end{definition}

That is, convergence in a Hausdorff space is unique.

Examples

Properties

In terms of lifting properties

The separation conditions $T_0$ to $T_4$ may equivalently be understood as lifting properties against certain maps of finite topological spaces, among others.

This is discussed at separation axioms in terms of lifting properties, to which we refer for further details. Here we just briefly indicate the corresponding lifting diagrams.

In the following diagrams, the relevant finite topological spaces are indicated explicitly by illustration of their underlying point set and their open subsets:

• points (elements) are denoted by $\bullet$ with subscripts indicating where the points map to;

• boxes are put around open subsets,

• an arrow $\bullet_u \to \bullet_c$ means that $\bullet_c$ is in the topological closure of $\bullet_u$.

In the lifting diagrams for $T_2-T_4$ below, an arrow out of the given topological space $X$ is a map that determines (classifies) a decomposition of $X$ into a union of subsets with properties indicated by the picture of the finite space.

Notice that the diagrams for $T_2$-$T_4$ below do not in themselves imply $T_1$.

\begin{proposition} (Lifting property encoding $T_0$) \linebreak The following lifting property in Top equivalently encodes the separation axiom $T_0$:

\begin{tikzcd} [column sep={between origins, 40pt}, row sep={between origins, 40pt}] {\boxed{\bullet_0\leftrightarrow \bullet_1}} \ar[rr, { \forall }] \ar[dd] && X \ar[dd,{ (T_0) }] \ \ \bullet_{0=1} \ar[rr] \ar[uurr, dashed, { \exists }] && \bullet \end{tikzcd}

\end{proposition}

\begin{proposition} (Lifting property encoding $T_1$) \linebreak The following lifting property in Top equivalently encodes the separation axiom $T_1$:

\begin{tikzcd} [column sep={between origins, 40pt}, row sep={between origins, 40pt}] {\boxed{ \overset{\boxed{\bullet_{0}}}{}\searrow\underset{\bullet_1}{} }} % {\boxed{ \boxed{ {\bullet_0} }\rightarrow{\bullet_1} }} \ar[rr, { \forall }] \ar[dd] && X \ar[dd,{ (T_1) }] \ \ \bullet_{0=1} \ar[rr] \ar[uurr, dashed, { \exists }] && \bullet \end{tikzcd} \end{proposition}

\begin{proposition} (Lifting property encoding $T_2$) \linebreak The following lifting property in Top equivalently encodes the separation axiom $T_2$:

\begin{tikzcd} [column sep={between origins, 80pt}, row sep={between origins, 40pt}] \boxed{{\boxed{\bullet_x}, \boxed{\bullet_y}}} \ar[rr, { }] \ar[dd, hook, { \forall }{left}, ,{ (T_2) }{right}] && \boxed{ \overset{ \boxed{ \boxed{\bullet_x} \;\; \, \;\; \boxed{\bullet_y}} }{ \underset{ \bullet_X } { \searrow \;\, \swarrow } } } \ar[dd] \ \ X \ar[rr] \ar[uurr, dashed, { \exists }] && {\boxed{\bullet_{x=X=y}}} \end{tikzcd} \end{proposition}

\begin{proposition} (Lifting property encoding $T_3$) \linebreak The following lifting property in Top equivalently encodes the separation axiom $T_3$:

\begin{tikzcd} [column sep={between origins, 60pt}, row sep={between origins, 40pt}] {\boxed{\bullet_x} } \ar[dd,{ \forall }{left}, { (T_3) }{right}] \ar[rr] && { \boxed{ \boxed{ \overset{\boxed{\bullet_x}}{} \searrow \underset{\bullet_X}{} \swarrow \overset{\boxed{\bullet_U}}{} }!!!!!!! {\,\,\,\,\,\,\searrow\underset{\bullet_F}{} } } } \ar[dd] \ \ X \ar[rr] \ar[uurr, dashed, { \exists }] && {\boxed{ \overset{\boxed{\bullet_{x=X=U}}}{}\searrow\underset{\bullet_F}{} }} \end{tikzcd} \end{proposition}

\begin{proposition} (Lifting property encoding $T_4$) \linebreak The following lifting property in Top equivalently encodes the separation axiom $T_4$:

\begin{tikzcd} [column sep={between origins, 60pt}, row sep={between origins, 40pt}] \varnothing \ar[dd, { (T_4) }{right}] \ar[rr] && \boxed{ \boxed{\underset{\bullet_x}{}\swarrow \,\,\,\,\,\,\,} !!!!!!!! \,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\, \,\,\,\,\,} !!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!! !!!!!!!! !!!! \boxed{ \boxed{ \overset{\boxed{\bullet_u}}{} \searrow \underset{\bullet_X}{} \swarrow \boxed{ \overset{\boxed{\bullet_v}}{} !!!!!!! {\,\,\,\,\,\,\searrow\underset{\bullet_y}{} } }!!!!!!!%%%%%%%% !!!!!} %\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,} \ar[dd] \ \ X \ar[rr] \ar[uurr, dashed, { \exists }] && { \boxed{ \underset{\bullet_x}{}{\swarrow}
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, %\,\, %\,\,\,\,\,\,\, } !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! %!!%!!!!!!! \boxed{\overset{\boxed{\bullet_{U=X=V}}}{} {\searrow} \underset{\bullet_y}{} } } \end{tikzcd} \end{proposition}

Hausdorff reflection

There are various ways to see the existence and to construct the Hausdorff reflection. The following is maybe the quickest way to see the existence, even though it leaves the actual construction rather implicit.

Some readers may find the following a more direct way of constructing the Hausdorff reflection:

(vanMunster 14, section 4)

Monadicity

The topology on a compact Hausdorff space is given precisely by the (existent because compact, unique because Hausdorff) limit of each ultrafilter on the space. Accordingly, compact Hausdorff topological spaces are (perhaps surprisingly) described by a (large) algebraic theory. In fact, the category of compact Hausdorff spaces is monadic (over Set); the monad in question maps each set to the set ultrafilters on it. (The results of this paragraph require the ultrafilter theorem, a weak form of the axiom of choice; see ultrafilter monad.)

A compact Hausdorff locale (or space) is necessarily regular; a regular locale (or $T_0$ space) is necessarily Hausdorff. Accordingly, locale theory usually speaks of ‘compact regular’ locales instead of ‘compact Hausdorff’ locales, since the definition of regularity is easier and more natural. Then a version of the previous paragraph works for compact regular locales without the ultrafilter theorem, and indeed constructively over any topos.

Sobriety

Using classical logic (but not in constructive logic) every Hausdorff space is a sober topological space: Hausdorff implies sober.

Relation to compact spaces

Dense subspaces and Epimorphisms

\begin{proposition} \label{DenseSubspaceInclusionsAreEpimorphismsAmongHausdorffSpaces}

In the category of Hausdorff topological spaces (with continuous functions between them), the inclusion of a dense subspace

is an epimorphism.

\end{proposition}

Here is a proof in the language of category theory:

\begin{proof}

We have to show that for $(f,g)$ any pair of parallel morphisms out of $X$

into a Hausdorff space $Y$, the equality $f \circ i = g \circ i$ implies $f = g$. Equivalently, that $f \circ i = g \circ i$ implies that $1_X \colon X \to X$ is the equalizer of the pair $(f, g)$. But the equalizer $E \to X$ is formed by taking a pullback of the diagonal map $\Delta \colon Y \to Y \times Y$ along $(f, g) \colon X \to Y \times Y$:

Since $Y$ is Hausdorff, the subset $\Delta: Y \to Y \times Y$ is closed, and the pullback $E \hookrightarrow X$ of this closed subset along the continuous map $h = (f, g)$, which is $E = h^{-1}(\Delta)$, is also closed. Since $E$ is a closed subset of $X$ and contains a dense subspace $i: A \hookrightarrow X$, it must be all of $X$ (as a subset of itself). \end{proof}

Note, incidentally, that $X$ itself doesn’t have to be Hausdorff for the argument to go through.

Alternatively, here is a proof in the language of basic topology:

\begin{proof}

We have to show that for $(f,g)$ any pair of parallel morphisms out of $X$

into a Hausdorff space $Y$, the equality $f \circ i = g \circ i$ implies $f = g$. With classical logic we may equivalently show the contrapositive: That $f \neq g$ implies $f \circ i \neq g \circ i$.

So assume that $f \neq g$. This means that there exists $x \in X$ with $f(x) \neq g(x)$. Since $Y$ is Hausdorff, there exist then disjoint open neighbourhoods $O_{f(x)},\;O_{g(x)} \subset Y$, i.e. $f(x) \in O_{f(x)}$ and $g(x) \in O_{g(x)}$ with $O_{f(x)} \cap O_{g(x)} = \varnothing$.

But their preimages must intersect at least in $x \in f^{-1}\big( O_{f(x)} \big) \cap g^{-1}\big( O_{g(x)} \big)$. Since this intersection is an open subset (as preimages of open subsets are open by definition of continuous functions, and since finite intersections of open subsets are open by the definition of topological spaces) there exists a point $a \in A$ with $i(a) \in f^{-1}\big( O_{f(x)} \big) \cap g^{-1}\big( O_{g(x)} \big)$ (by definition of dense subset). But since then $f(i(a)) \in O_{f(x)}$ and $g(i(a)) \in O_{g(x)}$ while $O_{f(x)}$ is disjoint from $O_{g(x)}$, it follows that $f(i(a)) \neq g(i(a))$. This means that $f \circ i \neq g \circ i$. \end{proof}

\linebreak

In fact the converse holds: any epimorphism in the category of Hausdorff spaces has dense image (e.g. LL 18).

Related notions

Arguably, the desire to make spaces Hausdorff ($T_2$) in analysis is really a desire to make them $T_0$; nearly every space that arises in analysis is at least regular, and a regular $T_0$ space must be Hausdorff. Forcing a space to be $T_0$ is like forcing a category to be skeletal; indeed, forcing a preorder to be a partial order is a special case of both (see specialisation topology for how). It may be nice to assume, when working with a particular space, that it is $T_0$ but not to assume, when working with a particular underlying set, that every topology on it is $T_0$.

Whatever one thinks of that, there is a non-$T_0$ version of Hausdorff space, an $R_1$ space. (The symbol here comes from being a weak version of a regular space; in general a $T_i$ space is precisely both $R_{i-1}$ and $T_0$). This is also called a preregular space (in HAF) and a reciprocal space (in convergence theory).

There is also a notion of sequentially Hausdorff space:

Some forms of predicative mathematics find this concept more useful. Hausdorffness implies sequential Hausdorffness, but the converse is false even for sequential spaces (although it is true for first-countable spaces).

The reader can now easily define a sequentially $R_1$ space.

Beyond topological spaces

Hausdorff locales

The most obvious definition for a locale $X$ to be Hausdorff is that its diagonal $X\to X\times X$ is a closed (and hence proper) inclusion. However, if $X$ is a sober space regarded as a locale, this might not coincide with the condition for $X$ to be Hausdorff as a space, since the Cartesian product $X\times X$ in the category Loc of locales might not coincide with the product in the category Top of topological spaces (the Tychonoff product). But it does coincide if $X$ is a locally compact locale, so in that case the two notions of Hausdorff are the same.

Separated toposes and schemes

This notion of a Hausdorff locale is a special case of that of Hausdorff topos in topos theory. This also is formally similar to notions such as a separated scheme etc. The corresponding relative notion (over an arbitrary base topos) is that of separated geometric morphism. For schemes see separated morphism of schemes.

In constructive mathematics

In constructive mathematics, the Hausdorff notion multifurcates further, due to the variety of possible meanings of closed subspace. If we ask the diagonal to be weakly closed, then in the spatial case, this means that it contains all its limit points, giving Definition 2 above. But if we ask the diagonal to be strongly closed, i.e. the complement of an open set, then in the spatial case this means that there is a tight inequality $\ne$ (the exterior of $=$) relative to which Definition 1 holds. (We use $\ne$ twice in that definition: in the hypothesis that $x \ne y$ and in the conclusion that $x' \ne y'$.)

It is natural to call these conditions weakly Hausdorff and strongly Hausdorff, but one should be aware of terminological clashes: in classical mathematics there is a different notion of a weak Hausdorff space, whereas (strong) Hausdorffness for locales has by some authors been called “strongly Hausdorff” only to contrast it with Hausdorffness for spaces.

As a simple example, consider a discrete space $X$ regarded as a locale. Since it is locally compact, the locale product $X\times X$ coincides with the space product (a theorem that is valid constructively); but nevertheless we have:

1. The diagonal $X\to X\times X$ always has an open complement.
2. Definition 2 above always holds, since $\{x\}$ and $\{y\}$ are neighborhoods of $x$ and $y$, and if they intersect then $x=y$. That is, $X$ is spatially weakly Hausdorff.
3. The diagonal $X\to X\times X$ is the complement of an open subset (i.e. $X$ is spatially strongly Hausdorff) if and only if equality in $X$ is stable under double negation, in other words if $X$ admits a tight inequality relation.
4. The locale diagonal $\Delta:X\to X\times X$ is a closed sublocale (i.e. $X$ is localically strongly Hausdorff) if and only if $X$ has decidable equality. For closedness of $\Delta$ means that $\Delta_\ast(U\cup \Delta^\ast(V)) \subseteq \Delta_\ast(U) \cup V$ for any $U\in O(X)$ and $V\in O(X\times X)$, where $x\in \Delta^\ast(V)$ means $(x,x)\in V$, while $(x,y)\in \Delta_\ast(U)$ means $(x=y)\to (x\in U)$. Now let $U = \emptyset$ and $V = \{ (x,x) \mid x\in X \}$. Then $(x,y) \in \Delta_\ast(U\cup \Delta^\ast(V))$ means $(x=y) \to (\bot \vee (x=x))$, which is a tautology; while $(x,y) \in \Delta_\ast(U) \cup V$ means $((x=y)\to \bot) \vee (x=y)$, i.e. $\neg(x=y) \vee (x=y)$.
5. I don’t know what it means for $X$ to be localically weakly Hausdorff. (Weak closure in locales is very inexplicit.)

In particular, the statement “all discrete locales are localically strongly Hausdorff” is equivalent to excluded middle.

However, non-discrete spaces can constructively be localically strongly Hausdorff without having decidable equality. For instance, any regular space is also regular as a locale, and hence localically strongly Hausdorff. We can also say:

Note that the apartness $\#$ need not be tight, and in particular $X$ need not be spatially Hausdorff. In particular, if $X$ might not even be $T_0$: since localic Hausdorffness is (of course) only a property of the open-set lattice, it only “sees” the sobrification and in particular the $T_0$ quotient (Kolmogorov quotient). However, this is all that can go wrong: if $\neg(x\# y)$, then by $\#$-openness every open set containing $x$ must also contain $y$ and vice versa, so if $X$ is $T_0$ then $x=y$ and $\#$ is tight.

If the locale product $X\times X$ does not coincide with the spatial product, then the “only if” direction of the above proof still works, if we define $W_\#$ to be the open part of the locale product $X\times X$ given by $W_\# = \bigvee \{ U\otimes V \mid U\cap V = \emptyset \}$. A different proof is to recall that by this theorem, an apartness relation is the same as a (strongly) closed equivalence relation on a discrete locale, and the quotient of such an equivalence relation is the $\#$-topology. Thus, if $X$ is localically strongly Hausdorff, its diagonal is a closed equivalence relation, which yields by pullback a closed equivalence relation on the discrete locale $X_d$ on the same set of points. This is the kernel pair of the canonical surjection $X_d \to X$, and hence its quotient (the $\#$-topology) maps to $X$, i.e. refines the topology of $X$.

History

First introduced by Hausdorff in Grundzüge der Mengenlehre, Hausdorff spaces were the original concept of a topological space. Later the Hausdorffness condition was dropped, apparently first by Kuratowski in 1920s.

Related concepts

• non-Hausdorff topological space

• locally Hausdorff topological space

• Kolmogorov topological space

• nice topological space

References

The notion is due to

• Felix Hausdorff, Grundzüge der Mengenlehre, Leipzig: Veit (1914) [ISBN:978-0-8284-0061-9] Reprinted by Chelsea Publishing Company (1944, 1949, 1965) [[ark:/13960/t2891gn8g](https://archive.org/details/grundzgedermen00hausuoft/page/n5/mode/2up)]

See also

• Wikipedia, Hausdorff space

and the references at topology.

A detailed discussion of Hausdorff reflection is in

• Bart van Munster, The Hausdorff quotient, 2014 (pdf)

On epimorphisms of Hausdorff spaces:

• Jérôme Lapuyade-Lahorgue, The epimorphisms of the category Haus are exactly the image-dense morphisms (arXiv:1810.00778)

Comments on the relation to topos theory:

• S. Niefield, A note on the locally Hausdorff property, Cahiers TGDC (1983) (numdam)